Contraction coefficients have been studied for several \(f\) -divergences, e.g.,
\(\eta _{\mathsf {TV}}\)  for total variation distance was studied in [1]}, [2]}, [3]}, \(\eta _{\mathsf {KL}}\)  for \(\mathsf {KL}\) -divergence was studied in [4]}, [5]}, [6]}, [7]}, [8]}, [9]}, and \(\eta _{\chi ^2}\)  for \(\chi ^2\) -divergence was studied in [3]}, [9]}, [12]}.
In particular, Dobrushin [1]} showed that \(\eta _{\mathsf {TV}}\) 
has a remarkably simple two-point characterization \(\eta _\mathsf {TV}({\mathsf {K}}) = \sup _{x_1,x_2\in {\mathcal {X}}} \mathsf {TV}({\mathsf {K}}(\cdot |x_1), {\mathsf {K}}(\cdot |x_2))\) .
